×

Nonwandering operators in Banach space. (English) Zbl 1100.47005

Let \(X\) be a Banach space and \(L(X)\) be the set of all bounded linear operators over \(X\). Let \(T\in L(X)\) and assume the existence of a closed subspace \(E\subset X\) with hyperbolic structure: \(E=E^u\oplus E^s\), \(TE^u=E^u\) and \(TE^s=E^s\) (where \(E^u\) and \(E^s\) are closed subspaces of \(E\)). The operator \(T\) is said to be a nonwandering operator related to \(E\) when the set Per\((T)\) of periodic points of \(T\) is dense in \(E\), there exist constants \(t\) (\(0<t<1\)) and \(c\) (\(c>0\)) such that \(\| T^ku\| \geq ct^{-k}\| u\| \) for all \(u\in E^u\) and all \(k\in{\mathbb N}\) and, in addition, \(\| T^ks\| \leq ct^k\| s\| \) for all \(s\in E^s\) and all \(k\in{\mathbb N}\).
In the paper under review, the authors state the existence of nonwandering operators on every infinite-dimensional separable Banach sequence space with unconditional basis, and some physical background examples are exposed.
Some properties of these operators, including the spectra decomposition of invertible nonwandering operators and its local structural stability, are stated. In particular, the authors show that the spectrum of any invertible nonwandering operator does not meet the unit circle and that any integer power of a nonwandering operator also enjoys the same property.
The concept of nonwandering operator is strongly related to that of hypercyclicity, that is, to the existence of a dense orbit.

MSC:

47A16 Cyclic vectors, hypercyclic and chaotic operators
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
PDFBibTeX XMLCite
Full Text: DOI EuDML